Construction of supercharacter theories of finite groups

Much can be learned about a finite group from its character table, but sometimes that table can be difficult to compute. Supercharacter theories are generalizations of character theory, defined by P. Diaconis and I.M. Isaacs in [8], in which certain (possibly reducible) characters called supercharacters take the place of the irreducible characters, and a certain coarser partition of the group takes the place of the conjugacy classes. In particular, if K is a partition of a finite group G, there may exist a compatible partition X of the irreducible characters of G, along with a character χX for everyX ∈ X with the elements of X as its irreducible constituents, so that each χX is constant on each K ∈ K and |X | = |K|. If every irreducible character is a constituent of some χX , then the ordered pair (X ,K) is called a supercharacter theory. We present five new ways to construct new supercharacter theories out of supercharacter theories already known to exist, including a direct product, a lattice-theoretic join, two products over normal subgroups, and a duality for supercharacter theories of abelian groups.